Study On The Influence Of Subgroup Properties On The Structure Of Finite Groups

Let G be a finite group,π(G)denotes the set of all primes dividing |G|.Letπ?π(G),a subgroup H of G is said to be a cπ-normal subgroup in G if there exists a normal subgroup K of G containing HG such that G=HK and H∩K/HG is a π’-group.And σ=｛σi{| i∈I} be a partition of the set of all primes P,that is,P=∪i∈Iσi and σi∩σj=? for all i≠j.The natural numbers n and m are called σ-coprime if σ(n)∩σ(m)=?.The group G is said to be:σ-primary if G is a σi-group for some i∈I;σ-soluble if either G=1 or every chief factor of G isσ-primary.A subgroup H of G is called σ-subnormal in G if there is a subgroup chain H=H0≤…≤Ht=G such that either Hi-1 is normal in Hi or Hi/(Hi-1)Hi isσ-primary for all i∈{1,…,t}.The research content of this thesis can be divided into four chapters,and the specific contents are as follows:The first chapter is the introduction,which mainly introduces the research background,the source of the problem and some basic concepts.In the second chapter,we mainly study the influence of cπ-normal subgroups on the structure of finite groups,including solvability,supersolvability,etc.In the third chapter,we mainly study σ-solvable groups and σ-subnormal subgroups of group G,and give some theorems to determine whether finite groups areσ-solvable groups and some theorems to determine whether subgroups of G are σsubnormal subgroups.In the fourth chapter,we summarize the work of this paper and introduce the work that can be further studied.